Fluid Mechanics, Thermodynamics of Turbomachinery

Transcription

1 Fluid Mechanics, Thermodynamics of Turbomachinery S.L. Dixon, B.Eng., PH.D. Senior Fellow at the University of Liverpool FOURTH EDITION in SI/METRIC UNITS

2 Fluid Mechanics, Thermodynamics of Turbomachinery FOURTH EDITION in SI/METRIC UNITS

3 In memory of Avril and baby Paul

4 Fluid Mechanics, Thermodynamics of Turbomachinery S. L. Dixon, B.Eng., Ph.D. Senior Fellow at the University of Liverpool FOURTH EDITION in SI/METRIC UNITS

5 Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA A division of Reed Educational and Professional Publishing Ltd A member of the Reed Elsevier plc group First published by Pergamon Press Ltd 1966 Second edition 1975 Third edition 1978 Reprinted 1979, 1982 (twice), 1984, 1986, 1989, 1992, 1995 Fourth edition 1998 S.L. Dixon 1978, 1998 All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a license issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress Typeset by Laser Words, Madras, India Printed and bound in

6 Contents PREFACE TO FOURTH EDITION ix PREFACE TO THIRD EDITION xi ACKNOWLEDGEMENTS xiii LIST OF SYMBOLS xv 1. Introduction: Dimensional Analysis: Similitude 1 Definition of a turbomachine 1 Units and dimensions 3 Dimensional analysis and performance laws 4 Incompressible fluid analysis 6 Performance characteristics 7 Variable geometry turbomachines 9 Specific speed 10 Cavitation 12 Compressible gas flow relations 15 Compressible fluid analysis 16 The inherent unsteadiness of the flow within turbomachines 20 References 21 Problems Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 23 Introduction 23 The equation of continuity 23 The first law of thermodynamics internal energy 24 The momentum equation Newton s second law of motion 25 The second law of thermodynamics entropy 29 Definitions of efficiency 30 Small stage or polytropic efficiency 35 Nozzle efficiency 41 Diffusers 43 References 53 Problems 53

7 vi Contents 3. Two-dimensional Cascades 55 Introduction 55 Cascade nomenclature 56 Analysis of cascade forces 57 Energy losses 59 Lift and drag 59 Circulation and lift 61 Efficiency of a compressor cascade 62 Performance of two-dimensional cascades 63 The cascade wind tunnel 63 Cascade test results 65 Compressor cascade performance 68 Turbine cascade performance 70 Compressor cascade correlations 71 Fan blade design (McKenzie) 80 Turbine cascade correlation (Ainley) 81 Comparison of the profile loss in a cascade and in a turbine stage 86 Optimum space-chord ratio of turbine blades (Zweifel) 87 References 88 Problems Axial-flow Turbines: Two-dimensional Theory 93 Introduction 93 Velocity diagrams of the axial turbine stage 93 Thermodynamics of the axial turbine stage 94 Stage losses and efficiency 96 Soderberg s correlation 97 Types of axial turbine design 99 Stage reaction 101 Diffusion within blade rows 103 Choice of reaction and effect on efficiency 107 Design point efficiency of a turbine stage 108 Maximum total-to-static efficiency of a reversible turbine stage 112 Stresses in turbine rotor blades 114 Turbine flow characteristics 120 Flow characteristics of a multistage turbine 122 The Wells turbine 124 References 132 Problems Axial-flow Compressors and Fans 137 Introduction 137 Two-dimensional analysis of the compressor stage 138 Velocity diagrams of the compressor stage 140 Thermodynamics of the compressor stage 141

8 Contents vii Stage loss relationships and efficiency 142 Reaction ratio 143 Choice of reaction 143 Stage loading 144 Simplified off-design performance 145 Stage pressure rise 147 Pressure ratio of a multistage compressor 148 Estimation of compressor stage efficiency 149 Stall and surge phenomena in compressors 154 Control of flow instabilities 159 Axial-flow ducted fans 160 Blade element theory 162 Blade element efficiency 163 Lift coefficient of a fan aerofoil 164 References 165 Problems Three-dimensional Flows in Axial Turbomachines 169 Introduction 169 Theory of radial equilibrium 169 The indirect problem 171 The direct problem 179 Compressible flow through a fixed blade row 180 Constant specific mass flow 181 Off-design performance of a stage 183 Free-vortex turbine stage 184 Actuator disc approach 186 Blade row interaction effects 190 Computer-aided methods of solving the through-flow problem 191 Secondary flows 193 References 195 Problems Centrifugal Pumps, Fans and Compressors 199 Introduction 199 Some definitions 200 Theoretical analysis of a centrifugal compressor 202 Inlet casing 203 Impeller 203 Conservation of rothalpy 204 Diffuser 205 Inlet velocity limitations 205 Optimum design of a pump inlet 206 Optimum design of a centrifugal compressor inlet 208 Slip factor 213 Head increase of a centrifugal pump 218

9 viii Contents Performance of centrifugal compressors 219 The diffuser system 227 Choking in a compressor stage 230 References 232 Problems Radial Flow Gas Turbines 236 Introduction 236 Types of inward flow radial turbine 237 Thermodynamics of the 90 deg IFR turbine 239 Basic design of the rotor 241 Nominal design point efficiency 242 Mach number relations 246 Loss coefficients in 90 deg IFR turbines 247 Optimum efficiency considerations 248 Criterion for minimum number of blades 253 Design considerations for rotor exit 256 Incidence losses 260 Significance and application of specific speed 263 Optimum design selection of 90 deg IFR turbines 266 Clearance and windage losses 269 Pressure ratio limits of the 90 deg IFR turbine 269 Cooled 90 deg IFR turbines 271 References 272 Problems Hydraulic Turbines 277 Introduction 277 Hydraulic turbines 278 The Pelton turbine 281 Reaction turbines 290 The Francis turbine 290 The Kaplan turbine 296 Effect of size on turbomachine efficiency 299 Cavitation 301 References 305 Problems 306 Bibliography 309 Appendix 1. Conversion of British and US Units to SI Units 310 Appendix 2. Answers to Problems 311 Index 315

10 Preface to the Fourth Edition It is now twenty years since the third edition of this book was published and in that period many advances have been made to the art and science of turbomachinery design. Knowledge of the flow processes within turbomachines has increased dramatically resulting in the appearance of new and innovative designs. Some of the long-standing, apparently intractable, problems such as surge and rotating stall have begun to yield to new methods of control. New types of flow machine have made their appearance (e.g. the Wells turbine and the axi-fuge compressor) and some changes have been made to established design procedures. Much attention is now being given to blade and flow passage design using computational fluid dynamics (CFD) and this must eventually bring forth further design and flow efficiency improvements. However, the fundamentals do not change and this book is still concerned with the basics of the subject as well as looking at new ideas. The book was originally perceived as a text for students taking an Honours degree in engineering which included turbomachines as well as assisting those undertaking more advanced postgraduate courses in the subject. The book was written for engineers rather than mathematicians. Much stress is laid on physical concepts rather than mathematics and the use of specialised mathematical techniques is mostly kept to a minimum. The book should continue to be of use to engineers in industry and technological establishments, especially as brief reviews are included on many important aspects of turbomachinery giving pointers to more advanced sources of information. For those looking towards the wider reaches of the subject area some interesting reading is contained in the bibliography. It might be of interest to know that the third edition was published in four languages. A fairly large number of additions and extensions have been included in the book from the new material mentioned as well as tidying up various sections no longer to my liking. Additions include some details of a new method of fan blade design, the determination of the design point efficiency of a turbine stage, sections on centrifugal stresses in turbine blades and blade cooling, control of flow instabilities in axial-flow compressors, design of the Wells turbine, consideration of rothalpy conservation in impellers (and rotors), defining and calculating the optimum efficiency of inward flow turbines and comparison with the nominal design. A number of extensions of existing topics have been included such as updating and extending the treatment and application of diffuser research, effect of prerotation of the flow in centrifugal compressors and the use of backward swept vanes on their performance, also changes in the design philosophy concerning the blading of axial-flow compressors. The original chapter on radial flow turbines has been split into two chapters; one dealing with radial gas turbines with some new extensions and the other on hydraulic turbines. In a world striving for a greener future it was felt that there would now be more than just a little interest in hydraulic turbines. It is a subject that is usually included in many mechanical engineering courses. This chapter includes a few new ideas which could be of some interest.

11 x Preface to the Fourth Edition A large number of illustrative examples have been included in the text and many new problems have been added at the end of most chapters (answers are given at the end of the book)! It is planned to publish a new supplementary text called Solutions Manual, hopefully, shortly after this present text book is due to appear, giving the complete and detailed solutions of the unsolved problems. S. Lawrence Dixon

12 Preface to Third Edition Several modifications have been incorporated into the text in the light of recent advances in some aspects of the subject. Further information on the interesting phenomenon of cavitation has been included and a new section on the optimum design of a pump inlet together with a worked example have been added which take into account recently published data on cavitation limitations. The chapter on three-dimensional flows in axial turbomachines has been extended; in particular the section concerning the constant specific mass flow design of a turbine nozzle has been clarified and now includes the flow equations for a following rotor row. Some minor alterations on the definition of blade shapes were needed so I have taken the opportunity of including a simplified version of the parabolic arc camber line as used for some low camber blading. Despite careful proof reading a number of errors still managed to elude me in the second edition. I am most grateful to those readers who have detected errors and communicated with me about them. In order to assist the reader I have (at last) added a list of symbols used in the text. S.L.D. xi

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14 Acknowledgements The author is indebted to a number of people and manufacturing organisations for their help and support; in particular the following are thanked: Professor W. A. Woods, formerly of Queen Mary College, University of London and a former colleague at the University of Liverpool for his encouragement of the idea of a fourth edition of this book as well as providing papers and suggestions for some new items to be included. Professor F. A. Lyman of Syracuse University, New York and Professor J. Moore of Virginia Polytechnic Institute and State University, Virginia, for their helpful correspondence and ideas concerning the vexed question of the conservation of rothalpy in turbomachines. Dr Y. R. Mayhew is thanked for supplying me with generous amounts of material on units and dimensions and the latest state of play on SI Units. Thanks are also given to the following organisations for providing me with illustrative material for use in the book, product information and, in one case, useful background historical information: Sulzer Hydro of Zurich, Switzerland; Rolls-Royce of Derby, England; Voith Hydro Inc., Pennsylvania; and Kvaerner Energy, Norway. Last, but by no means least, to my wife Rose, whose quiet patience and support enabled this new edition to be prepared.

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16 List of Symbols A area a sonic velocity, position of maximum camber b passage width, maximum camber C f tangential force coefficient C L,C D lift and drag coefficients C p specific heat at constant pressure, pressure coefficient, pressure rise coefficient C pi ideal pressure rise coefficient C v specific heat at constant volume C X,C Y axial and tangential force coefficients c absolute velocity c o spouting velocity D drag force, diameter D eq equivalent diffusion ratio D h hydraulic mean diameter E, e energy, specific energy F c centrifugal force in blade f acceleration, friction factor g gravitational acceleration H head, blade height H E effective head H f head loss fue to friction H G gross head H S net positive suction head (NPSH) h specific enthalpy I rothalpy i incidence angle K, k constants K N nozzle velocity coefficient L lift force, length of diffuser wall l blade chord length, pipe length M Mach number m mass, molecular weight N rotational speed, axial length of diffuser N S specific speed (rev) N SP power specific speed (rev) N SS suction specific speed (rev) n number of stages, polytropic index p pressure

17 xvi Fluid Mechanics, Thermodynamics of Turbomachinery p a p v atmospheric pressure vapour pressure Q heat transfer, volume flow rate q dryness fraction R reaction, specific gas constant Re Reynolds number R H reheat factor R o universal gas constant r radius S entropy, power ratio s blade pitch, specific entropy T temperature t time, thickness U blade speed, internal energy u specific internal energy V, v volume, specific volume W work transfer 1W specific work transfer w relative velocity X axial force x, y, z Cartesian coordinate directions Y tangential force, actual tangential blade load per unit span Y id ideal tangential blade load per unit span Y k tip clearance loss coefficient Y p profile loss coefficient Y S net secondary loss coefficient Z number of blades, Ainley blade loading parameter absolute flow angle ˇ relative flow angle 0 circulation ratio of specific heats υ deviation angle ε fluid deflection angle, cooling effectiveness enthalpy loss coefficient, total pressure loss coefficient efficiency 2 minimum opening at cascade exit blade camber angle, wake momentum thickness profile loss coefficient dynamic viscosity kinematic viscosity, blade stagger angle, velocity ratio density slip factor, solidity b blade cavitation coefficient c Thoma s coefficient, centrifugal stress torque

18 flow coefficient, velocity ratio 9 stage loading factor speed of rotation (rad/s) S specific speed (rad) SP power specific speed (rad) SS suction specific speed (rad) ω vorticity ω stagnation pressure loss coefficient List of Symbols xvii Subscripts av average c compressor, critical D diffuser e exit h hydraulic, hub i inlet, impeller id ideal is isentropic m mean, meridional, mechanical, material N nozzle n normal component o stagnation property, overall p polytropic, constant pressure R reversible process, rotor r radial rel relative s isentropic, stall condition ss stage isentropic t turbine, tip, transverse v velocity x, y, z cartesian coordinate components tangential Superscript Ð time rate of change - average 0 blade angle (as distinct from flow angle) * nominal condition

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20 CHAPTER 1 Introduction: Dimensional Analysis: Similitude If you have known one you have known all. (TERENCE, Phormio.) Definition of a turbomachine We classify as turbomachines all those devices in which energy is transferred either to, or from, a continuously flowing fluid by the dynamic action of one or more moving blade rows. The word turbo or turbinis is of Latin origin and implies that which spins or whirls around. Essentially, a rotating blade row, a rotor or an impeller changes the stagnation enthalpy of the fluid moving through it by either doing positive or negative work, depending upon the effect required of the machine. These enthalpy changes are intimately linked with the pressure changes occurring simulataneously in the fluid. The definition of a turbomachine as stated above, is rather too general for the purposes of this book as it embraces open turbomachines such as propellers, wind turbines and unshrouded fans, all of which influence the state of a not readily quantifiable flow of a fluid. The subject fluid mechanics, thermodynamics of turbomachinery, therefore, is limited to machines enclosed by a closely fitting casing or shroud through which a readily measurable quantity of fluid passes in unit time. The subject of open turbomachines is covered by the classic text of Glauert (1959) or by Duncan et al. (1970), the elementary treatment of propellers by general fluid mechanics textbooks such as Streeter and Wylie (1979) or Massey (1979), and the important, still developing subject of wind turbines, by Freris (1990). Two main categories of turbomachine are identified: firstly, those which absorb power to increase the fluid pressure or head (ducted fans, compressors and pumps); secondly, those that produce power by expanding fluid to a lower pressure or head (hydraulic, steam and gas turbines). Figure 1.1 shows, in a simple diagrammatic form, a selection of the many different varieties of turbomachine encountered in practice. The reason that so many different types of either pump (compressor) or turbine are in use is because of the almost infinite range of service requirements. Generally speaking, for a given set of operating requirements there is one type of pump or turbine best suited to provide optimum conditions of operation. This point is discussed more fully in the section of this chapter concerned with specific speed. Turbomachines are further categorised according to the nature of the flow path through the passages of the rotor. When the path of the through-flow is wholly or mainly parallel to the axis of rotation, the device is termed an axial flow turbomachine (e.g. 1

21 2 Fluid Mechanics, Thermodynamics of Turbomachinery FIG Diagrammatic form of various types of turbomachine. Figure 1.1(a) and (e)). When the path of the through-flow is wholly or mainly in a plane perpendicular to the rotation axis, the device is termed a radial flow turbomachine (e.g. Figure 1.1(c)). More detailed sketches of radial flow machines are given in Figures 7.1, 7.2, 8.2 and 8.3. Mixed flow turbomachines are widely used. The term mixed flow in this context refers to the direction of the through-flow at rotor outlet when both radial and axial velocity components are present in significant amounts. Figure 1.1(b) shows a mixed flow pump and Figure 1.1(d) a mixed flow hydraulic turbine. One further category should be mentioned. All turbomachines can be classified as either impulse or reaction machines according to whether pressure changes are

22 Introduction: Dimensional Analysis: Similitude 3 absent or present respectively in the flow through the rotor. In an impulse machine all the pressure change takes place in one or more nozzles, the fluid being directed onto the rotor. The Pelton wheel, Figure 1.1(f), is an example of an impulse turbine. The main purpose of this book is to examine, through the laws of fluid mechanics and thermodynamics, the means by which the energy transfer is achieved in the chief types of turbomachine, together with the differing behaviour of individual types in operation. Methods of analysing the flow processes differ depending upon the geometrical configuration of the machine, on whether the fluid can be regarded as incompressible or not, and whether the machine absorbs or produces work. As far as possible, a unified treatment is adopted so that machines having similar configurations and function are considered together. Units and dimensions The International System of Units, SI (le Système International d Unités) is a unified self-consistent system of measurement units based on the MKS (metre kilogram second) system. It is a simple, logical system based upon decimal relationships between units making it easy to use. The most recent detailed description of SI has been published in 1986 by HMSO. For an explanation of the relationship between, and use of, physical quantities, units and numerical values see Quantities, Units and Symbols, published by The Royal Society (1975) or refer to ISO 31/ Great Britain was the first of the English-speaking countries to begin, in the 1960s, the long process of abandoning the old Imperial System of Units in favour of the International System of Units, and was soon followed by Canada, Australia, New Zealand and South Africa. In the USA a ten year voluntary plan of conversion to SI units was commenced in In 1975 US President Ford signed the Metric Conversion Act which coordinated the metrication of units, but did so without specifying a schedule of conversion. Industries heavily involved in international trade (cars, aircraft, food and drink) have, however, been quick to change to SI for obvious economic reasons, but others have been reluctant to change. SI has now become established as the only system of units used for teaching engineering in colleges, schools and universities in most industrialised countries throughout the world. The Imperial System was derived arbitrarily and has no consistent numerical base, making it confusing and difficult to learn. In this book all numerical problems involving units are performed in metric units as this is more convenient than attempting to use a mixture of the two systems. However, it is recognised that some problems exist as a result of the conversion to SI units. One of these is that many valuable papers and texts written prior to 1969 contain data in the old system of units and would need converting to SI units. A brief summary of the conversion factors between the more frequently used Imperial units and SI units is given in Appendix 1 of this book. Some SI units The SI basic units used in fluid mechanics and thermodynamics are the metre (m), kilogram (kg), second (s) and thermodynamic temperature (K). All the other units used in this book are derived from these basic units. The unit of force is the

23 4 Fluid Mechanics, Thermodynamics of Turbomachinery newton (N), defined as that force which, when applied to a mass of 1 kilogram, gives an acceleration to the mass of 1 m/s 2. The recommended unit of pressure is the pascal (Pa) which is the pressure produced by a force of 1 newton uniformly distributed over an area of 1 square metre. Several other units of pressure are in widespread use, however, foremost of these being the bar. Much basic data concerning properties of substances (steam and gas tables, charts, etc.) have been prepared in SI units with pressure given in bars and it is acknowledged that this alternative unit of pressure will continue to be used for some time as a matter of expediency. It is noted that 1 bar equals 10 5 Pa (i.e N/m 2 ), roughly the pressure of the atmosphere at sea level, and is perhaps an inconveniently large unit for pressure in the field of turbomachinery anyway! In this book the convenient size of the kilopascal (kpa) is found to be the most useful multiple of the recommended unit and is extensively used in most calculations and examples. In SI the units of all forms of energy are the same as for work. The unit of energy is the joule (J) which is the work done when a force of 1 newton is displaced through a distance of 1 metre in the direction of the force, e.g. kinetic energy ( 1 2 mc2 ) has the dimensions kg ð m 2 /s 2 ; however, 1 kg D 1Ns 2 /m from the definition of the newton given above. Hence, the units of kinetic energy must be Nm D J upon substituting dimensions. The watt (W) is the unit of power; when 1 watt is applied for 1 second to a system the input of energy to that system is 1 joule (i.e. 1 J). The hertz (Hz) is the number of repetitions of a regular occurrence in 1 second. Instead of writing c/s for cycles/sec, Hz is used instead. The unit of thermodynamic temperature is the kelvin (K), written without the sign, and is the fraction 1/ of the thermodynamic temperature of the triple point of water. The degree celsius ( C) is equal to the unit kelvin. Zero on the celsius scale is the temperature of the ice point ( K). Specific heat capacity, or simply specific heat, is expressed as J/kg K or as J/kg C. Dynamic viscosity, dimensions ML 1 T 1, has the SI units of pascal seconds, i.e. M LT kg m.s D N.s2 D Pa s. m. 2 s Hydraulic engineers find it convenient to express pressure in terms of head of a liquid. The static pressure at any point in a liquid at rest is, relative to the pressure acting on the free surface, proportional to the vertical distance of the free surface above that point. The head H is simply the height of a column of the liquid which can be supported by this pressure. If is the mass density (kg/m 3 ) and g the local gravitational acceleration (m/s 2 ), then the static pressure p (relative to atmospheric pressure) is p D gh, where H is in metres and p is in pascals (or N/m 2 ). This is left for the student to verify as a simple exercise. Dimensional analysis and performance laws The widest comprehension of the general behaviour of all turbomachines is, without doubt, obtained from dimensional analysis. This is the formal procedure whereby the group of variables representing some physical situation is reduced

24 Introduction: Dimensional Analysis: Similitude 5 into a smaller number of dimensionless groups. When the number of independent variables is not too great, dimensional analysis enables experimental relations between variables to be found with the greatest economy of effort. Dimensional analysis applied to turbomachines has two further important uses: (a) prediction of a prototype s performance from tests conducted on a scale model (similitude); (b) determination of the most suitable type of machine, on the basis of maximum efficiency, for a specified range of head, speed and flow rate. Several methods of constructing non-dimensional groups have been described by Douglas et al. (1995) and by Shames (1992) among other authors. The subject of dimensional analysis was made simple and much more interesting by Edward Taylor (1974) in his comprehensive account of the subject. It is assumed here that the basic techniques of forming non-dimensional groups have already been acquired by the student. Adopting the simple approach of elementary thermodynamics, an imaginary envelope (called a control surface) of fixed shape, position and orientation is drawn around the turbomachine (Figure 1.2). Across this boundary, fluid flows steadily, entering at station 1 and leaving at station 2. As well as the flow of fluid there is a flow of work across the control surface, transmitted by the shaft either to, or from, the machine. For the present all details of the flow within the machine can be ignored and only externally observed features such as shaft speed, flow rate, torque and change in fluid properties across the machine need be considered. To be specific, let the turbomachine be a pump (although the analysis could apply to other classes of turbomachine) driven by an electric motor. The speed of rotation N, can be adjusted by altering the current to the motor; the volume flow rate Q, can be independently adjusted by means of a throttle valve. For fixed values of the set Q and N, all other variables such as torque, head H, are thereby established. The choice of Q and N as control variables is clearly arbitrary and any other pair of independent variables such as and H could equally well have been chosen. The important point to recognise is, that there are for this pump, two control variables. If the fluid flowing is changed for another of different density, and viscosity, the performance of the machine will be affected. Note, also, that for a turbomachine handling compressible fluids, other fluid properties are important and are discussed later. So far we have considered only one particular turbomachine, namely a pump of a given size. To extend the range of this discussion, the effect of the geometric FIG Turbomachine considered as a control volume.

25 6 Fluid Mechanics, Thermodynamics of Turbomachinery variables on the performance must now be included. The size of machine is characterised by the impeller diameter D, and the shape can be expressed by a number of length ratios, l 1 /D, l 2 /D, etc. Incompressible fluid analysis The performance of a turbomachine can now be expressed in terms of the control variables, geometric variables and fluid properties. For the hydraulic pump it is convenient to regard the net energy transfer gh, the efficiency, and power supplied P, as dependent variables and to write the three functional relationships as ( gh D f 1 Q, N, D,,, l 1 D, l ) 2 D,..., 1.1a ( D f 2 Q, N, D,,, l 1 D, l ) 2 D,..., 1.1b ( P D f 3 Q, N, D,,, l 1 D, l ) 2 D,..., 1.1c By the procedure of dimensional analysis using the three primary dimensions, mass, length and time, or alternatively, using three of the independent variables we can form the dimensionless groups. The latter, more direct procedure, requires that the variables selected,, N, D, do not of themselves form a dimensionless group. The selection of, N, D as common factors avoids the appearance of special fluid terms (e.g., Q) in more than one group and allows gh, and P to be made explicit. Hence the three relationships reduce to the following easily verified forms. Energy transfer coefficient, sometimes called head coefficient D gh ( Q ND D f 2 4 ND, ND2 3, l 1 D, l ) 2 D,..., 1.2a ( Q D f 5 ND, ND2 3, l 1 D, l ) 2 D, b Power coefficient ( Q ND, ND2 3, l 1 D, l ) 2 D,.... (1.2c) OP D P N 3 D D f 5 6 The non-dimensional group Q/ ND 3 is a volumetric flow coefficient and ND 2 / is a form of Reynolds number, Re. In axial flow turbomachines, an alternative to Q/ ND 3 which is frequently used is the velocity (or flow) coefficient D c x /U where U is blade tip speed and c x the average axial velocity. Since and then Q D c x ð flow area / c x D 2 U / ND. Q ND 3 / c x U.

26 Introduction: Dimensional Analysis: Similitude 7 Because of the large number of independent groups of variables on the right-hand side of eqns. (1.2), those relationships are virtually worthless unless certain terms can be discarded. In a family of geometrically similar machines l 1 /D, l 2 /D are constant and may be eliminated forthwith. The kinematic viscosity, D / is very small in turbomachines handling water and, although speed, expressed by ND, is low the Reynolds number is correspondingly high. Experiments confirm that effects of Reynolds number on the performance are small and may be ignored in a first approximation. The functional relationships for geometrically similar hydraulic turbomachines are then, D f 4 [Q/ ND 3 ] D f 5 [Q/ ND 3 ] OP D f 6 [Q/ ND 3 ]. 1.3a 1.3b 1.3c This is as far as the reasoning of dimensional analysis alone can be taken; the actual form of the functions f 4, f 5 and f 6 must be ascertained by experiment. One relation between,, and OP may be immediately stated. For a pump the net hydraulic power, P N equals QgH which is the minimum shaft power required in the absence of all losses. No real process of power conversion is free of losses and the actual shaft power P must be larger than P N. We define pump efficiency (more precise definitions of efficiency are stated in Chapter 2) D P N /P D QgH/P. Therefore P D 1 ( ) Q gh ND 3 ND 2 N3 D 5. (1.4) Thus f 6 may be derived from f 4 and f 5 since OP D /. For a turbine the net hydraulic power P N supplied is greater than the actual shaft power delivered by the machine and the efficiency D P/P N. This can be rewritten as OP D by reasoning similar to the above considerations. Performance characteristics The operating condition of a turbomachine will be dynamically similar at two different rotational speeds if all fluid velocities at corresponding points within the machine are in the same direction and proportional to the blade speed. If two points, one on each of two different head flow characteristics, represent dynamically similar operation of the machine, then the non-dimensional groups of the variables involved, ignoring Reynolds number effects, may be expected to have the same numerical value for both points. On this basis, non-dimensional presentation of performance data has the important practical advantage of collapsing into virtually a single curve, results that would otherwise require a multiplicity of curves if plotted dimensionally. Evidence in support of the foregoing assertion is provided in Figure 1.3 which shows experimental results obtained by the author (at the University of Liverpool) on a simple centrifugal laboratory pump. Within the normal operating range of this pump, 0.03 < Q/ ND 3 <0.06, very little systematic scatter is apparent which

27 8 Fluid Mechanics, Thermodynamics of Turbomachinery might be associated with a Reynolds number effect, for the range of speeds 2500 N 5000 rev/min. For smaller flows, Q/ ND 3 <0.025, the flow became unsteady and the manometer readings of uncertain accuracy but, nevertheless, dynamically similar conditions still appear to hold true. Examining the results at high flow rates one is struck by a marked systematic deviation away from the single-curve law at increasing speed. This effect is due to cavitation, a high speed phenomenon of hydraulic machines caused by the release of vapour bubbles at low pressures, which is discussed later in this chapter. It will be clear at this stage that under cavitating flow conditions, dynamical similarity is not possible. FIG Dimensionless head-volume characteristic of a centrifugal pump. FIG Extrapolation of characteristic curves for dynamically similar conditions at N D 3500 rev/min.

28 Introduction: Dimensional Analysis: Similitude 9 The non-dimensional results shown in Figure 1.3 have, of course, been obtained for a particular pump. They would also be approximately valid for a range of different pump sizes so long as all these pumps are geometrically similar and cavitation is absent. Thus, neglecting any change in performance due to change in Reynolds number, the dynamically similar results in Figure 1.3 can be applied to predicting the dimensional performance of a given pump for a series of required speeds. Figure 1.4 shows such a dimensional presentation. It will be clear from the above discussion that the locus of dynamically similar points in the H Q field lies on a parabola since H varies as N 2 and Q varies as N. Variable geometry turbomachines The efficiency of a fixed geometry machine, ignoring Reynolds number effects, is a unique function of flow coefficient. Such a dependence is shown by line (b) in Figure 1.5. Clearly, off-design operation of such a machine is grossly inefficient and designers sometimes resort to a variable geometry machine in order to obtain a better match with changing flow conditions. Figure 1.6 shows a sectional sketch of a mixed-flow pump in which the impeller vane angles may be varied during pump operation. (A similar arrangement is used in Kaplan turbines, Figure 1.1.) Movement of the vanes is implemented by cams driven from a servomotor. In some very large installations involving many thousands of kilowatts and where operating FIG Different efficiency curves for a given machine obtained with various blade settings. FIG Mixed-flow pump incorporating mechanism for adjusting blade setting.

29 10 Fluid Mechanics, Thermodynamics of Turbomachinery conditions fluctuate, sophisticated systems of control may incorporate an electronic computer. The lines (a) and (c) in Figure 1.5 show the efficiency curves at other blade settings. Each of these curves represents, in a sense, a different constant geometry machine. For such a variable geometry pump the desired operating line intersects the points of maximum efficiency of each of these curves. Introducing the additional variable ˇ into eqn. (1.3) to represent the setting of the vanes, we can write D f 1, ˇ ; D f 2, ˇ. (1.5) Alternatively, with ˇ D f 3, D f 4,, ˇ can be eliminated to give a new functional dependence ( ) Q D f 5, D f 5 ND, gh (1.6) 3 N 2 D 2 Thus, efficiency in a variable geometry pump is a function of both flow coefficient and energy transfer coefficient. Specific speed The pump or hydraulic turbine designer is often faced with the basic problem of deciding what type of turbomachine will be the best choice for a given duty. Usually the designer will be provided with some preliminary design data such as the head H, the volume flow rate Q and the rotational speed N when a pump design is under consideration. When a turbine preliminary design is being considered the parameters normally specified are the shaft power P, the head at turbine entry H and the rotational speed N. A non-dimensional parameter called the specific speed, N s, referred to and conceptualised as the shape number, is often used to facilitate the choice of the most appropriate machine. This new parameter is derived from the non-dimensional groups defined in eqn. (1.3) in such a way that the characteristic diameter D of the turbomachine is eliminated. The value of N s gives the designer a guide to the type of machine that will provide the normal requirement of high efficiency at the design condition. For any one hydraulic turbomachine with fixed geometry there is a unique relationship between efficiency and flow coefficient if Reynolds number effects are negligible and cavitation absent. As is suggested by any one of the curves in Figure 1.5, the efficiency rises to a maximum value as the flow coefficient is increased and then gradually falls with further increase in. This optimum efficiency D max, is used to identify a unique value D 1 and corresponding unique values of D 1 and OP D OP 1. Thus, Q ND D 3 1 D constant, 1.7a gh N 2 D D 2 1 D constant, 1.7b P N 3 D D OP 5 1 D constant. 1.7c

30 Introduction: Dimensional Analysis: Similitude 11 It is a simple matter to combine any pair of these expressions in such a way as to eliminate the diameter. For a pump the customary way of eliminating D is to divide 1/2 1 by 3/4 1. Thus N s D 1/2 1 3/4 1 D NQ1/2, (1.8) gh 3/4 where N s is called the specific speed. The term specific speed is justified to the extent that N s is directly proportional to N. In the case of a turbine the power specific speed N sp, is more useful and is defined by, P N sp D O 1/2 1 5/4 1 D N P/ 1/2 gh 5/4 (1.9) Both eqns. (1.8) and (1.9) are dimensionless. It is always safer and less confusing to calculate specific speed in one or other of these forms rather than dropping the factors g and which would make the equations dimensional and any values of specific speed obtained using them would then depend upon the choice of the units employed. The dimensionless form of N s (and N sp ) is the only one used in this book. Another point arises from the fact that the rotational speed, N, is expressed in the units of revolutions per unit of time so that although N s is dimensionless, numerical values of specific speed need to be thought of as revs. Alternative versions of eqns. (1.8) and (1.9) in radians are also in common use and are written s D Q1/2 gh 3/4, 1.8a sp D p P/ gh. 1.9a 5/4 There is a simple connection between N s and N sp (and between s and sp ). By dividing eqn. (1.9) by eqn. (1.8) we obtain N sp N s D N P/ 1/2 gh 3/4 gh 5/4 NQ 1/2 D ( ) P 1/2. gqh From the definition of hydraulic efficiency, for a pump we obtain: N sp N s D sp s D 1 p, (1.9b) and, for a turbine we obtain: N sp N s D sp s D p. (1.9c) Remembering that specific speed, as defined above, is at the point of maximum efficiency of a turbomachine, it becomes a parameter of great importance in selecting the type of machine required for a given duty. The maximum efficiency condition replaces the condition of geometric similarity, so that any alteration in specific

31 12 Fluid Mechanics, Thermodynamics of Turbomachinery FIG Range of pump impellers of equal inlet area. speed implies that the machine design changes. Broadly speaking, each different class of machine has its optimum efficiency within its own fairly narrow range of specific speed. For a pump, eqn. (1.8) indicates, for constant speed N, that N s is increased by an increase in Q and decreased by an increase in H. From eqn. (1.7b) it is observed that H, at a constant speed N, increased with impeller diameter D. Consequently, to increase N s the entry area must be made large and/or the maximum impeller diameter small. Figure 1.7 shows a range of pump impellers varying from the axialflow type, through mixed flow to a centrifugal- or radial-flow type. The size of each inlet is such that they all handle the same volume flow Q. Likewise, the head developed by each impeller (of different diameter D) is made equal by adjusting the speed of rotation N. Since Q and H are constant, then N s varies with N alone. The most noticeable feature of this comparison is the large change in size with specific speed. Since a higher specific speed implies a smaller machine, for reasons of economy, it is desirable to select the highest possible specific speed consistent with good efficiency. Cavitation In selecting a hydraulic turbomachine for a given head H and capacity Q, itis clear from the definition of specific speed, eqn. (1.8), that the highest possible value of N s should be chosen because of the resulting reduction in size, weight and cost. On this basis a turbomachine could be made extremely small were it not for the corresponding increase in the fluid velocities. For machines handling liquids the lower limit of size is dictated by the phenomenon of cavitation. Cavitation is the boiling of a liquid at normal temperature when the static pressure is made sufficiently low. It may occur at the entry to pumps or at the exit from hydraulic turbines in the vicinity of the moving blades. The dynamic action of the blades causes the static pressure to reduce locally in a region which is already normally below atmospheric pressure and cavitation can commence. The phenomenon is accentuated by the presence of dissolved gases which are released with a reduction in pressure. For the purpose of illustration consider a centrifugal pump operating at constant speed and capacity. By steadily reducing the inlet pressure head a point is reached

32 Introduction: Dimensional Analysis: Similitude 13 when streams of small vapour bubbles appear within the liquid and close to solid surfaces. This is called cavitation inception and commences in the regions of lowest pressure. These bubbles are swept into regions of higher pressure where they collapse. This condensation occurs suddenly, the liquid surrounding the bubbles either hitting the walls or adjacent liquid. The pressure wave produced by bubble collapse (with a magnitude of the order 400 MPa) momentarily raises the pressure level in the vicinity and the action ceases. The cycle then repeats itself and the frequency may be as high as 25 khz (Shepherd 1956). The repeated action of bubbles collapsing near solid surfaces leads to the well-known cavitation erosion. The collapse of vapour cavities generates noise over a wide range of frequencies up to 1 MHz has been measured (Pearsall 1972) i.e. so-called white noise. Apparently it is the collapsing smaller bubbles which cause the higher frequency noise and the larger cavities the lower frequency noise. Noise measurement can be used as a means of detecting cavitation (Pearsall 1966/7). Pearsall and McNulty (1968) have shown experimentally that there is a relationship between cavitation noise levels and erosion damage on cylinders and concludes that a technique could be developed for predicting the occurrence of erosion. Up to this point no detectable deterioration in performance has occurred. However, with further reduction in inlet pressure, the bubbles increase both in size and number, coalescing into pockets of vapour which affects the whole field of flow. This growth of vapour cavities is usually accompanied by a sharp drop in pump performance as shown conclusively in Figure 1.3 (for the 5000 rev/min test data). It may seem surprising to learn that with this large change in bubble size, the solid surfaces are much less likely to be damaged than at inception of cavitation. The avoidance of cavitation inception in conventionally designed machines can be regarded as one of the essential tasks of both pump and turbine designers. However, in certain recent specialised applications pumps have been designed to operate under supercavitating conditions. Under these conditions large size vapour bubbles are formed but, bubble collapse takes place downstream of the impeller blades. An example of the specialised application of a supercavitating pump is the fuel pumps of rocket engines for space vehicles where size and mass must be kept low at all costs. Pearsall (1966) has shown that the supercavitating principle is most suitable for axial flow pumps of high specific speed and has suggested a design technique using methods similar to those employed for conventional pumps. Pearsall (1966) was one of the first to show that operating in the supercavitating regime was practicable for axial flow pumps and he proposed a design technique to enable this mode of operation to be used. A detailed description was later published (Pearsall 1973), and the cavitation performance was claimed to be much better than that of conventional pumps. Some further details are given in Chapter 7 of this book. Cavitation limits In theory cavitation commences in a liquid when the static pressure is reduced to the vapour pressure corresponding to the liquid s temperature. However, in practice, the physical state of the liquid will determine the pressure at which cavitation starts (Pearsall 1972). Dissolved gases come out of solution as the pressure is reduced forming gas cavities at pressures in excess of the vapour pressure. Vapour cavitation requires the presence of nuclei submicroscopic gas bubbles or solid non-wetted

33 14 Fluid Mechanics, Thermodynamics of Turbomachinery particles in sufficient numbers. It is an interesting fact that in the absence of such nuclei a liquid can withstand negative pressures (i.e. tensile stresses)! Perhaps the earliest demonstration of this phenomenon was that performed by Osborne Reynolds (1882) before a learned society. He showed how a column of mercury more than twice the height of the barometer could be (and was) supported by the internal cohesion (stress) of the liquid. More recently Ryley (1980) devised a simple centrifugal apparatus for students to test the tensile strength of both plain, untreated tap water in comparison with water that had been filtered and then de-aerated by boiling. Young (1989) gives an extensive literature list covering many aspects of cavitation including the tensile strength of liquids. At room temperature the theoretical tensile strength of water is quoted as being as high as 1000 atm (100 MPa)! Special pretreatment (i.e. rigorous filtration and pre-pressurization) of the liquid is required to obtain this state. In general the liquids flowing through turbomachines will contain some dust and dissolved gases and under these conditions negative pressure do not arise. A useful parameter is the available suction head at entry to a pump or at exit from a turbine. This is usually referred to as the net positive suction head, NPSH, defined as H s D p o p / g (1.10) where p o and p are the absolute stagnation and vapour pressures, respectively, at pump inlet or at turbine outlet. To take into account the effects of cavitation, the performance laws of a hydraulic turbomachine should include the additional independent variable H s. Ignoring the effects of Reynolds number, the performance laws of a constant geometry hydraulic turbomachine are then dependent on two groups of variable. Thus, the efficiency, D f, N ss (1.11) where the suction specific speed N ss D NQ 1/2 / gh s 3/4, determines the effect of cavitation, and D Q/ ND 3, as before. It is known from experiment that cavitation inception occurs for an almost constant value of N ss for all pumps (and, separately, for all turbines) designed to resist cavitation. This is because the blade sections at the inlet to these pumps are broadly similar (likewise, the exit blade sections of turbines are similar) and it is the shape of the low pressure passages which influences the onset of cavitation. Using the alternative definition of suction specific speed ss D Q 1/2 / gh s 1/2, where is the rotational speed in rad/s, Q is the volume flow in m 3 /s and gh s,is in m 2 /s 2, it has been shown empirically (Wislicehus 1947) that ss ' 3.0 (rad) for pumps, and (1.12a) ss ' 4.0 (rad) (1.12b) for turbines. Pearsall (1973) described a supercavitating pump with a cavitation performance much better that of conventional pumps. For this pump suction specific speeds, ss